2 Basic shapes of geometry
COMMENT. Geometry deals with our intuitions as to the physical space in which
we exist, with the properties of the shapes and sizes of bodies as mathematically
abstracted. It differs from set theory in that in geometry there are distinguished or
special subsets, and relations involving them. To start with we presuppose a moderate
knowledge of set theory, sufficient to deal with sets, relations and functions. From
Chapter 3 on we assume a knowledge of the real number system, and the elementary
algebra involved.
In this first chapter we introduce the plane, points, lines, natural orders on lines,
and open half-planes as primitive concepts, and in terms of these develop other spe-
cial types of geometrical sets.
2.1 Lines, segments and half-lines.....................
2.1.1 Plane,points,lines...........................
Primitive Terms. Assuming the terminology of sets, theplane, denoted byΠ,isa
universal set the elements of which are calledpoints. Certain subsets ofΠare called
(straight)lines. We denote byΛthe set of all these lines.
AXIOM A 1 .Each line is a proper non-empty subset ofΠ. For each set{A,B}of
two distinct points inΠ, there is a unique line inΛto which A and B both belong.|
We denote byABthe unique line to which distinct pointsAandBbelong, so that
A∈ABandB∈AB. It is an immediate consequence of Axiom A 1 thatAB=BA;that
ifCandDare distinct points and both belong to the lineAB,thenAB=CD;andthat
ifA,Bare distinct points, both on the linelandbothonthelinem,thenl=m.
Furthermore ifl,mare any two lines inΛ, then either
l∩m= 0 /,Geometry with Trigonometry
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